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A boundary driven generalized contact process with exchange of particles Hydrodynamics in infinite volume.pdf

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A boundary driven generalized contact process with exchange of particles Hydrodynamics in infinite volume.pdf

Available online at ScienceDirect Stochastic Processes and their Applications 127 (2017) 135–178 /locate/spa A boundary driven generalized contact process with exchange of particles: Hydrodynamics in infinite volume Kevin Kuocha, Mustapha Mourraguib,?, Ellen Saadac a Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, PO Box 407, 9700AK Groningen, The Netherlands b LMRS, UMR 6085, Universite′ de Rouen, Avenue de l’Universite′, BP.12, Technopo?le du Madrillet, F76801 Saint-E′tienne-du-Rouvray, France c CNRS, UMR 8145, Laboratoire MAP5, Universite′ Paris Descartes, Sorbonne Paris Cite′, 45 rue des Saints-Pe`res, 75270 Paris Cedex 06, France Received 1 October 2015; received in revised form 17 May 2016; accepted 5 June 2016 Available online 21 June 2016 Abstract We consider a two species process which evolves in a finite or infinite domain in contact with particle reservoirs at different densities, according to the superposition of a generalized contact process and a rapidstirring dynamics in the bulk of the domain, and a creation/annihilation mechanism at its boundaries. For this process, we study the law of large numbers for densities and current. The limiting equations are given by a system of non-linear reaction–diffusion equations with Dirichlet boundary conditions. ?c 2016 Elsevier B.V. All rights reserved. MSC: primary 82C22; secondary 60F10; 82C35 Keywords: Generalized contact process; Two species process; Hydrodynamic limit; Specific entropy; Stationary nonequilibrium states; Reservoirs; Infinite volume; Current ? Corresponding author. E-mail addresses: kevin.kuoch@ (K. Kuoch), mustapha.mourragui@univ-rouen.fr (M. Mourragui), Ellen.Saada@mi.parisdescartes.fr (E. Saada). /10.1016/j.spa.2016.06.004 0304-4149/?c 2016 Elsevier B.V. All rights reserved. 136 K. Kuoch et al. / Stochastic Processes and their Applications 127 (2017) 135–178 1. Introduction In this paper, we consider the evolution on a lattice of two types of popul